Background

In October 2012, the Center for Medicare Services (CMS) began implementing the Hospital Readmissions Reduction Program (HRRP). The HRRP financially penalizes hospitals with excess readmissions. CMS characterizes excess readmissions using a ratio of the hospital’s number of “predicted” 30-day readmissions for the given condition to the “expected” number of readmissions for the condition based on an average hospital with similar patients. Thus, a readmissions ratio greater than one indicates worse than average performance in readmissions and a readmissions ratio less than one indicates better than average performance. Currently, the HRRP focuses on readmission rates for 6 conditions: heart attack (AMI), heart failure (HF), pneumonia (PN), chronic obstructive pulmonary disease (COPD), elective hip or knee replacement (HIP-KNEE), and coronary artery bypass graft (CABG)

As noted in an issue brief from the Kaiser Family Foundation, national readmission rates have consitently fallen since 2012, suggesting hospitals and clinicians “may have adopted new, system-wide interventions soon after the HRRP was enacted”. However, the report notes that some types of hospitals are still more likely than others to be penalized under the HRRP. Specifically, the report notes hosepitals that are major teaching hospitals, in rural areas, and serve more low-income beneficiaries are more likely to be penalized and have higher penalities.

One hypothesis is that certain types of hospitals have fewer resources to successfully implement systemic changes that help reduce readmission rates, and therefore reduce penalizations under the HRRP. If this hypothesis were true, additional policies could be implemented to (1) adjust the calculation of the excess readmissions ratio to account for some of these differences, (2) adjust the HRRP penalties for certain hospital characteristics, and/or (3) support hospitals with fewer resources in pursuing changes to reduce readmission rates.

Using publicly available data from the CMS on the HRRP program and general hospital information, this analysis continues to explore hospital characteristics associated with higher excess readmission rates overall and for each of the 6 conditions separately.

Key Results

  1. Overall and condition specific excess readmission rate averages appear to vary between states.

  2. The variance in excess readmission rates differences between conditions.

  3. Overall and condition specific excess readmission rate distributions appear to vary with hospital ownership for some conditions. However, sample sizes for some ownership types are small, so we should be cautious in drawing inferences from these results.

  4. Presence of emergency services is not associated with a difference in excess readmission rates, except for the elective hip or knee replacement condition.

Possible Next Steps

  1. Expand exploratory analysis using additional/more granular data on hospital characeristics and zipcode level demographics for further exploration. Explore differences by combinations of hospital characteristics.

  2. Explore further the distributional differences in excess readmission ratio by hospital ownership. Where there is sufficient sample size, are the observed differences statistically significant?

  3. Explore further the difference in excess readmission ratio by presence of emergency services for the elective hip or knee replacement condition.

  4. If we want to estimate the average causal effect of the HRRP on hospital excess readmission ratio (or some other outcome), we could explore using an interrupted time series analysis.

Methods

Primary methods used were visualizations of readmissions mean and standard deviations by state, summary statistics, box plots, density plots, OLS regressions, and t-tests.

More specifically:

Results

Overall Excess Admissions Rates

First, I review some characteristics for the excess readmissions ratio across all conditions.

Measure ID Distributions

The overall mean of the readmission ratio for each condition is roughly equal to 1, as expected. The readmission ratio for each condition is roughly normally distributed, although the conditions have different sample sizes and variance.

counts <- dat %>% 
  group_by(measure_id) %>% 
  count()

readm_ratio_stats <- dat %>% 
  select(c(readm_ratio, measure_id)) %>%
  group_by(measure_id) %>% 
  summarise_all(c("mean", "median", "sd"), na.rm = TRUE) 

reduce(list(readm_ratio_stats, counts), left_join) %>%
  kable %>% 
  kable_styling(bootstrap_options = c("striped", "hover"))
measure_id mean median sd n
AMI 1.002086 1.00035 0.0617432 1998
CABG 1.000805 0.99350 0.0998617 967
COPD 1.001223 0.99660 0.0623454 2711
HF 1.001548 0.99920 0.0772992 2723
HIP-KNEE 1.006320 0.99225 0.1397141 2352
PN 1.002480 0.99680 0.0830292 2794
ggplot(dat, aes(readm_ratio, group = measure_id, color = measure_id)) + 
  geom_density()

Overall excess readmission ratios by state

The overall excress readmission ratios vary by state. This trend is also seen within each each condition, although those plots are not displayed here. As expected, hospitals with lower overall ratings tend to have higher excess admissions ratios.

dat_state_avgs <-
  dat %>% 
  group_by(state) %>% 
  summarise(readm_ratio_avg = mean(readm_ratio), 
            readm_ratio_sd = sd(readm_ratio), 
            hospital_rating_mean = mean(as.numeric(hospital_overall_rating), na.rm = T)) %>% 
  arrange(readm_ratio_avg)

readmit_state_plot(dat_state_avgs)

dat_measure_id_state_avgs <-
  dat %>% 
  group_by(measure_id, state) %>% 
  summarise(readm_ratio_avg = mean(readm_ratio), 
            readm_ratio_sd = sd(readm_ratio), 
            hospital_rating_mean = mean(as.numeric(hospital_overall_rating), na.rm = T))

readmit_state_plot(dat_measure_id_state_avgs %>% filter(measure_id == "AMI"))
readmit_state_plot(dat_measure_id_state_avgs %>% filter(measure_id == "CABG"))
readmit_state_plot(dat_measure_id_state_avgs %>% filter(measure_id == "COPD"))
readmit_state_plot(dat_measure_id_state_avgs %>% filter(measure_id == "HF"))
readmit_state_plot(dat_measure_id_state_avgs %>% filter(measure_id == "HIP-KNEE"))
readmit_state_plot(dat_measure_id_state_avgs %>% filter(measure_id == "PN"))

Correlations between factors used to compute hospital ratings

There are 7 factors, including the readmission ratio, used to calculate overall hospital ratings. They do not appear to be highly correlated, however, the general hospital information only includes data on if each of these factors is roughly equal to the national average, above the national average, or below the national average. More granular data on these factors might be more highly correlated.

# look at correlations between 7 factors used to compute hospital rating
# they aren't very highly correlated 
dat_subset <- dat %>% select(readm_ratio, 
                             effectiveness_of_care_national_comparison,
                             efficient_use_of_medical_imaging_national_comparison,
                             mortality_national_comparison, 
                             patient_experience_national_comparison, 
                             safety_of_care_national_comparison, 
                             timeliness_of_care_national_comparison)

dat_subset[ls(dat_subset)] <- lapply(dat_subset[ls(dat_subset)], as.numeric)
cormat <- round(cor(dat_subset, use = "pairwise.complete.obs"), 2)
upper_tri <- get_upper_tri(cormat)
melted_cormat <- melt(upper_tri, na.rm = TRUE)

ggplot(data = melted_cormat, aes(x=Var1, y=Var2, fill=value)) + 
  geom_tile() + 
  theme(axis.text.x = element_text(angle = 30, hjust = 1))

By condition

For each condition, summary stats, box plots, and density charts are created based on hospital ownership and presence of emergency services.

AMI

dat_AMI <- dat %>% filter(measure_id == "AMI")
  
readmit_stats(dat_AMI, quo(hospital_ownership))
hospital_ownership mean median sd n
Government - Federal 1.0008857 0.96690 0.0744747 7
Government - Hospital District or Authority 0.9936984 0.98940 0.0504780 122
Government - Local 0.9968754 0.99760 0.0605228 69
Government - State 1.0161630 1.01960 0.0539914 27
Physician 0.9827273 0.97750 0.0739384 11
Proprietary 1.0164091 1.01420 0.0626097 385
Voluntary non-profit - Church 0.9961557 0.99620 0.0634395 194
Voluntary non-profit - Other 0.9983356 0.99675 0.0530341 194
Voluntary non-profit - Private 0.9996471 0.99850 0.0632510 989
readmit_boxplot(dat_AMI, "hospital_ownership")

readmit_density(dat_AMI, "hospital_ownership")

mod <- lm(readm_ratio ~ hospital_ownership, data = dat_AMI)
stargazer(mod, type = "text")
## 
## =========================================================================================
##                                                                   Dependent variable:    
##                                                               ---------------------------
##                                                                       readm_ratio        
## -----------------------------------------------------------------------------------------
## hospital_ownershipGovernment - Hospital District or Authority           -0.007           
##                                                                         (0.024)          
##                                                                                          
## hospital_ownershipGovernment - Local                                    -0.004           
##                                                                         (0.024)          
##                                                                                          
## hospital_ownershipGovernment - State                                     0.015           
##                                                                         (0.026)          
##                                                                                          
## hospital_ownershipPhysician                                             -0.018           
##                                                                         (0.030)          
##                                                                                          
## hospital_ownershipProprietary                                            0.016           
##                                                                         (0.023)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Church                         -0.005           
##                                                                         (0.024)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Other                          -0.003           
##                                                                         (0.024)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Private                        -0.001           
##                                                                         (0.023)          
##                                                                                          
## Constant                                                               1.001***          
##                                                                         (0.023)          
##                                                                                          
## -----------------------------------------------------------------------------------------
## Observations                                                             1,998           
## R2                                                                       0.015           
## Adjusted R2                                                              0.011           
## Residual Std. Error                                                0.061 (df = 1989)     
## F Statistic                                                     3.792*** (df = 8; 1989)  
## =========================================================================================
## Note:                                                         *p<0.1; **p<0.05; ***p<0.01
readmit_stats(dat_AMI, quo(emergency_services))
emergency_services mean median sd n
FALSE 0.9966048 1.0002 0.0577425 83
TRUE 1.0023237 1.0004 0.0619140 1915
readmit_boxplot(dat_AMI, "emergency_services")

readmit_density(dat_AMI, "emergency_services")

readmit_emergency_services_ttest(dat_AMI)
## 
##  Welch Two Sample t-test
## 
## data:  emergency_services and no_emergency_services
## t = 0.88063, df = 90.366, p-value = 0.3809
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.007182029  0.018619701
## sample estimates:
## mean of x mean of y 
## 1.0023237 0.9966048

CABG

dat_CABG <- dat %>% filter(measure_id == "CABG")
  
readmit_stats(dat_CABG, quo(hospital_ownership))
hospital_ownership mean median sd n
Government - Federal 1.1355000 1.13550 0.0353553 2
Government - Hospital District or Authority 1.0077276 1.00475 0.1036978 58
Government - Local 1.0055500 1.01465 0.0819842 22
Government - State 0.9933118 0.99920 0.1033186 17
Physician 0.9355571 0.90390 0.0750370 7
Proprietary 1.0268116 1.01660 0.1049831 189
Voluntary non-profit - Church 0.9971072 0.99270 0.0917395 111
Voluntary non-profit - Other 0.9856333 0.99510 0.0949837 78
Voluntary non-profit - Private 0.9935319 0.98430 0.0991429 483
readmit_boxplot(dat_CABG, "hospital_ownership")

readmit_density(dat_CABG %>% filter(), "hospital_ownership")

mod <- lm(readm_ratio ~ hospital_ownership, data = dat_CABG)
stargazer(mod, type = "text")
## 
## =========================================================================================
##                                                                   Dependent variable:    
##                                                               ---------------------------
##                                                                       readm_ratio        
## -----------------------------------------------------------------------------------------
## hospital_ownershipGovernment - Hospital District or Authority           -0.128*          
##                                                                         (0.071)          
##                                                                                          
## hospital_ownershipGovernment - Local                                    -0.130*          
##                                                                         (0.073)          
##                                                                                          
## hospital_ownershipGovernment - State                                    -0.142*          
##                                                                         (0.074)          
##                                                                                          
## hospital_ownershipPhysician                                            -0.200**          
##                                                                         (0.079)          
##                                                                                          
## hospital_ownershipProprietary                                           -0.109           
##                                                                         (0.070)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Church                         -0.138*          
##                                                                         (0.071)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Other                         -0.150**          
##                                                                         (0.071)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Private                       -0.142**          
##                                                                         (0.070)          
##                                                                                          
## Constant                                                               1.136***          
##                                                                         (0.070)          
##                                                                                          
## -----------------------------------------------------------------------------------------
## Observations                                                              967            
## R2                                                                       0.025           
## Adjusted R2                                                              0.017           
## Residual Std. Error                                                0.099 (df = 958)      
## F Statistic                                                     3.101*** (df = 8; 958)   
## =========================================================================================
## Note:                                                         *p<0.1; **p<0.05; ***p<0.01
readmit_stats(dat_CABG, quo(emergency_services))
emergency_services mean median sd n
FALSE 1.0196000 1.00515 0.1210953 40
TRUE 0.9999936 0.99340 0.0988413 927
readmit_boxplot(dat_CABG,  "emergency_services")

readmit_density(dat_CABG, "emergency_services")

readmit_emergency_services_ttest(dat_CABG)
## 
##  Welch Two Sample t-test
## 
## data:  emergency_services and no_emergency_services
## t = -1.0096, df = 41.273, p-value = 0.3186
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.05881820  0.01960547
## sample estimates:
## mean of x mean of y 
## 0.9999936 1.0196000

COPD

dat_COPD <- dat %>% filter(measure_id == "COPD")
  
readmit_stats(dat_COPD, quo(hospital_ownership))
hospital_ownership mean median sd n
Government - Federal 1.0067667 1.01450 0.0485975 12
Government - Hospital District or Authority 0.9899368 0.98720 0.0552764 231
Government - Local 1.0011496 1.00270 0.0596856 139
Government - State 0.9966853 0.99755 0.0515801 34
Physician 0.9966000 1.00815 0.0649841 14
Proprietary 1.0120134 1.00580 0.0622038 524
Tribal 0.9947000 0.99380 0.0406575 3
Voluntary non-profit - Church 0.9940424 0.98630 0.0623524 231
Voluntary non-profit - Other 1.0005862 0.99680 0.0629137 275
Voluntary non-profit - Private 1.0003967 0.99495 0.0637009 1248
readmit_boxplot(dat_COPD, "hospital_ownership")

readmit_density(dat_COPD, "hospital_ownership")

mod <- lm(readm_ratio ~ hospital_ownership, data = dat_COPD)
stargazer(mod, type = "text")
## 
## =========================================================================================
##                                                                   Dependent variable:    
##                                                               ---------------------------
##                                                                       readm_ratio        
## -----------------------------------------------------------------------------------------
## hospital_ownershipGovernment - Hospital District or Authority           -0.017           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipGovernment - Local                                    -0.006           
##                                                                         (0.019)          
##                                                                                          
## hospital_ownershipGovernment - State                                    -0.010           
##                                                                         (0.021)          
##                                                                                          
## hospital_ownershipPhysician                                             -0.010           
##                                                                         (0.024)          
##                                                                                          
## hospital_ownershipProprietary                                            0.005           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipTribal                                                -0.012           
##                                                                         (0.040)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Church                         -0.013           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Other                          -0.006           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Private                        -0.006           
##                                                                         (0.018)          
##                                                                                          
## Constant                                                               1.007***          
##                                                                         (0.018)          
##                                                                                          
## -----------------------------------------------------------------------------------------
## Observations                                                             2,711           
## R2                                                                       0.010           
## Adjusted R2                                                              0.007           
## Residual Std. Error                                                0.062 (df = 2701)     
## F Statistic                                                     3.016*** (df = 9; 2701)  
## =========================================================================================
## Note:                                                         *p<0.1; **p<0.05; ***p<0.01
readmit_stats(dat_COPD, quo(emergency_services))
emergency_services mean median sd n
FALSE 0.9966717 0.9914 0.0519082 127
TRUE 1.0014466 0.9968 0.0628136 2584
readmit_boxplot(dat_COPD, "emergency_services")

readmit_density(dat_COPD, "emergency_services")

readmit_emergency_services_ttest(dat_COPD)
## 
##  Welch Two Sample t-test
## 
## data:  emergency_services and no_emergency_services
## t = 1.0013, df = 144.75, p-value = 0.3184
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.00465089  0.01420077
## sample estimates:
## mean of x mean of y 
## 1.0014466 0.9966717

HF

dat_HF <- dat %>% filter(measure_id == "HF")
  
readmit_stats(dat_HF, quo(hospital_ownership))
hospital_ownership mean median sd n
Government - Federal 1.0246357 1.01890 0.0498174 14
Government - Hospital District or Authority 0.9980645 0.99505 0.0681573 228
Government - Local 1.0054331 0.99150 0.0822802 139
Government - State 1.0160500 1.00865 0.0548997 34
Physician 0.9695800 0.98440 0.0907027 15
Proprietary 1.0254674 1.02050 0.0728490 527
Tribal 0.9859000 0.98590 0.0183848 2
Voluntary non-profit - Church 0.9811538 0.97105 0.0750814 234
Voluntary non-profit - Other 0.9926814 0.98170 0.0737623 279
Voluntary non-profit - Private 0.9972237 0.99410 0.0796563 1251
readmit_boxplot(dat_HF, "hospital_ownership")

readmit_density(dat_HF, "hospital_ownership")

mod <- lm(readm_ratio ~ hospital_ownership, data = dat_HF)
stargazer(mod, type = "text")
## 
## =========================================================================================
##                                                                   Dependent variable:    
##                                                               ---------------------------
##                                                                       readm_ratio        
## -----------------------------------------------------------------------------------------
## hospital_ownershipGovernment - Hospital District or Authority           -0.027           
##                                                                         (0.021)          
##                                                                                          
## hospital_ownershipGovernment - Local                                    -0.019           
##                                                                         (0.021)          
##                                                                                          
## hospital_ownershipGovernment - State                                    -0.009           
##                                                                         (0.024)          
##                                                                                          
## hospital_ownershipPhysician                                             -0.055*          
##                                                                         (0.028)          
##                                                                                          
## hospital_ownershipProprietary                                            0.001           
##                                                                         (0.021)          
##                                                                                          
## hospital_ownershipTribal                                                -0.039           
##                                                                         (0.058)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Church                        -0.043**          
##                                                                         (0.021)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Other                          -0.032           
##                                                                         (0.021)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Private                        -0.027           
##                                                                         (0.020)          
##                                                                                          
## Constant                                                               1.025***          
##                                                                         (0.020)          
##                                                                                          
## -----------------------------------------------------------------------------------------
## Observations                                                             2,723           
## R2                                                                       0.029           
## Adjusted R2                                                              0.026           
## Residual Std. Error                                                0.076 (df = 2713)     
## F Statistic                                                     9.156*** (df = 9; 2713)  
## =========================================================================================
## Note:                                                         *p<0.1; **p<0.05; ***p<0.01
readmit_stats(dat_HF, quo(emergency_services))
emergency_services mean median sd n
FALSE 1.003645 0.99875 0.0702471 128
TRUE 1.001445 0.99920 0.0776414 2595
readmit_boxplot(dat_HF, "emergency_services")

readmit_density(dat_HF, "emergency_services")

readmit_emergency_services_ttest(dat_HF)
## 
##  Welch Two Sample t-test
## 
## data:  emergency_services and no_emergency_services
## t = -0.34404, df = 142.74, p-value = 0.7313
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.01483745  0.01043833
## sample estimates:
## mean of x mean of y 
##  1.001445  1.003645

HIP-KNEE

dat_HIPKNEE <- dat %>% filter(measure_id == "HIP-KNEE")
  
readmit_stats(dat_HIPKNEE, quo(hospital_ownership))
hospital_ownership mean median sd n
Government - Federal 1.0055100 1.01030 0.0800398 10
Government - Hospital District or Authority 1.0246125 0.99050 0.1425493 160
Government - Local 0.9950409 0.98680 0.1315563 93
Government - State 1.0582692 1.02205 0.1688633 26
Physician 0.9543500 0.94665 0.1185621 50
Proprietary 1.0301494 1.01230 0.1432590 449
Voluntary non-profit - Church 0.9972621 0.98970 0.1447585 214
Voluntary non-profit - Other 0.9927648 0.98055 0.1274366 244
Voluntary non-profit - Private 1.0008272 0.99020 0.1391104 1106
readmit_boxplot(dat_HIPKNEE, "hospital_ownership")

readmit_density(dat_HIPKNEE, "hospital_ownership")

mod <- lm(readm_ratio ~ hospital_ownership, data = dat_HIPKNEE)
stargazer(mod, type = "text")
## 
## =========================================================================================
##                                                                   Dependent variable:    
##                                                               ---------------------------
##                                                                       readm_ratio        
## -----------------------------------------------------------------------------------------
## hospital_ownershipGovernment - Hospital District or Authority            0.019           
##                                                                         (0.045)          
##                                                                                          
## hospital_ownershipGovernment - Local                                    -0.010           
##                                                                         (0.046)          
##                                                                                          
## hospital_ownershipGovernment - State                                     0.053           
##                                                                         (0.052)          
##                                                                                          
## hospital_ownershipPhysician                                             -0.051           
##                                                                         (0.048)          
##                                                                                          
## hospital_ownershipProprietary                                            0.025           
##                                                                         (0.044)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Church                         -0.008           
##                                                                         (0.045)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Other                          -0.013           
##                                                                         (0.045)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Private                        -0.005           
##                                                                         (0.044)          
##                                                                                          
## Constant                                                               1.006***          
##                                                                         (0.044)          
##                                                                                          
## -----------------------------------------------------------------------------------------
## Observations                                                             2,352           
## R2                                                                       0.014           
## Adjusted R2                                                              0.010           
## Residual Std. Error                                                0.139 (df = 2343)     
## F Statistic                                                     4.020*** (df = 8; 2343)  
## =========================================================================================
## Note:                                                         *p<0.1; **p<0.05; ***p<0.01
readmit_stats(dat_HIPKNEE, quo(emergency_services))
emergency_services mean median sd n
FALSE 0.973566 0.97445 0.1225367 162
TRUE 1.008743 0.99360 0.1406236 2190
readmit_boxplot(dat_HIPKNEE, "emergency_services")

readmit_density(dat_HIPKNEE, "emergency_services")

readmit_emergency_services_ttest(dat_HIPKNEE)
## 
##  Welch Two Sample t-test
## 
## data:  emergency_services and no_emergency_services
## t = 3.4879, df = 193.76, p-value = 0.0006024
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.01528570 0.05506841
## sample estimates:
## mean of x mean of y 
##  1.008743  0.973566

PN

dat_PN <- dat %>% filter(measure_id == "PN")
  
readmit_stats(dat_PN, quo(hospital_ownership))
hospital_ownership mean median sd n
Government - Federal 0.9935609 0.98500 0.0727920 23
Government - Hospital District or Authority 0.9928753 0.98600 0.0770911 243
Government - Local 1.0017611 0.99420 0.0769157 149
Government - State 1.0077917 0.99235 0.0743617 36
Physician 0.9640571 0.96260 0.0689033 14
Proprietary 1.0173490 1.01720 0.0804306 541
Tribal 0.9685250 0.97370 0.0564518 4
Voluntary non-profit - Church 0.9888502 0.97510 0.0916850 235
Voluntary non-profit - Other 1.0003759 0.99420 0.0785838 282
Voluntary non-profit - Private 1.0015969 0.99580 0.0850036 1267
readmit_boxplot(dat_PN, "hospital_ownership")

readmit_density(dat_PN, "hospital_ownership")

mod <- lm(readm_ratio ~ hospital_ownership, data = dat_PN)
stargazer(mod, type = "text")
## 
## =========================================================================================
##                                                                   Dependent variable:    
##                                                               ---------------------------
##                                                                       readm_ratio        
## -----------------------------------------------------------------------------------------
## hospital_ownershipGovernment - Hospital District or Authority           -0.001           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipGovernment - Local                                     0.008           
##                                                                         (0.019)          
##                                                                                          
## hospital_ownershipGovernment - State                                     0.014           
##                                                                         (0.022)          
##                                                                                          
## hospital_ownershipPhysician                                             -0.030           
##                                                                         (0.028)          
##                                                                                          
## hospital_ownershipProprietary                                            0.024           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipTribal                                                -0.025           
##                                                                         (0.045)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Church                         -0.005           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Other                           0.007           
##                                                                         (0.018)          
##                                                                                          
## hospital_ownershipVoluntary non-profit - Private                         0.008           
##                                                                         (0.017)          
##                                                                                          
## Constant                                                               0.994***          
##                                                                         (0.017)          
##                                                                                          
## -----------------------------------------------------------------------------------------
## Observations                                                             2,794           
## R2                                                                       0.011           
## Adjusted R2                                                              0.008           
## Residual Std. Error                                                0.083 (df = 2784)     
## F Statistic                                                     3.511*** (df = 9; 2784)  
## =========================================================================================
## Note:                                                         *p<0.1; **p<0.05; ***p<0.01
readmit_stats(dat_PN, quo(emergency_services))
emergency_services mean median sd n
FALSE 1.003285 0.9991 0.0655729 139
TRUE 1.002438 0.9965 0.0838528 2655
readmit_boxplot(dat_PN, "emergency_services")

readmit_density(dat_PN, "emergency_services")

readmit_emergency_services_ttest(dat_PN)
## 
##  Welch Two Sample t-test
## 
## data:  emergency_services and no_emergency_services
## t = -0.14615, df = 162.58, p-value = 0.884
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.01229016  0.01059623
## sample estimates:
## mean of x mean of y 
##  1.002438  1.003285